Wilcoxon Test for Paired Differences

In order to compare two matched samples we have two major possibilities: the t test for matched pairs if the differences of the pairs are normally distributed, and the signed rank test of Wilcoxon if the differences are not normally distributed.

The Wilcoxon test can be used to check whether the differences of matched pairs are distributed symmetrically around the median (provided that the median of the differences is zero). If the null hypothesis has to be rejected then either the medians are not equal, or the samples origin from two different distributions.

In order to perform the test one has first to calculate the pairwise differences between the N observations of the two groups. Pairs whose difference is zero will be discarded. The remaining M differences have to be sorted and ranked according to their absolute values. The ranks of tied differences have to be averaged. Next, the ranks of all positive and of all negative differences are summed up. The smaller of the two sums gives the test statistic W.

The null hypothesis has to be rejected if the statistic W is less than the critical limit W_α. The critical limits for two-tailed tests are listed in the following table (taken from McCornack 1965 ):

M	α=0.1	α=0.05%	α=0.02%	α=0.01%	α=0.001%
6	2	0	0	0	0
7	3	2	0	0	0
8	5	3	1	0	0
9	8	5	3	1	0
10	10	8	5	3	0
11	13	10	7	5	0
12	17	13	9	7	1
13	21	17	12	9	2
14	25	21	15	12	4
15	30	25	19	15	6
16	35	29	23	19	8
17	41	34	27	23	11
18	47	40	32	27	14
19	53	46	37	32	18
20	60	52	43	37	21
21	67	58	49	42	25
22	75	65	55	48	30
23	83	73	62	54	35
24	91	81	69	61	40
25	100	89	76	68	45
26	110	98	84	75	51
27	119	107	92	83	57
28	130	116	101	91	64
29	140	126	110	100	71
30	151	137	120	109	78
31	163	147	130	118	86
32	175	159	140	128	94
33	187	170	151	138	102
34	200	182	162	148	111
35	213	195	173	159	120
36	227	208	185	171	130
37	241	221	198	182	140
38	256	235	211	194	150
39	271	249	224	207	161
40	286	264	238	220	172
41	302	279	252	233	183
42	319	294	266	247	195
43	336	310	281	261	207
44	353	327	296	276	220
45	371	343	312	291	233
46	389	361	328	307	246
47	407	378	345	322	260
48	426	396	362	339	274
49	446	415	379	355	289
50	466	434	397	373	304
51	486	453	416	390	319
52	507	473	434	408	335
53	529	494	454	427	351
54	550	514	473	445	368
55	573	536	493	465	385
56	595	557	514	484	402
57	618	579	535	504	420
58	642	602	556	525	438
59	666	625	578	546	457
60	690	648	600	567	476
61	715	672	623	589	495
62	741	697	646	611	515
63	767	721	669	634	535
64	793	747	693	657	556
65	820	772	718	681	577
66	847	798	742	705	599
67	875	825	768	729	621
68	903	852	793	754	643
69	931	879	819	779	666
70	960	907	846	805	689
71	990	936	873	831	712
72	1020	964	901	858	736
73	1050	994	928	884	761
74	1081	1023	957	912	786
75	1112	1053	986	940	811
76	1144	1084	1015	968	836
77	1176	1115	1044	997	862
78	1209	1147	1075	1026	889
79	1242	1179	1105	1056	916
80	1276	1211	1136	1086	943
81	1310	1244	1168	1116	971
82	1345	1277	1200	1147	999
83	1380	1311	1232	1178	1028
84	1415	1345	1265	1210	1057
85	1451	1380	1298	1242	1086
86	1487	1415	1332	1275	1116
87	1524	1451	1366	1308	1146
88	1561	1487	1400	1342	1177
89	1599	1523	1435	1376	1208
90	1638	1560	1471	1410	1240
91	1676	1597	1507	1445	1271
92	1715	1635	1543	1480	1304
93	1755	1674	1580	1516	1337
94	1795	1712	1617	1552	1370
95	1836	1752	1655	1589	1404
96	1877	1791	1693	1626	1438
97	1918	1832	1731	1664	1472
98	1960	1872	1770	1702	1507
99	2003	1913	1810	1740	1543
100	2045	1955	1850	1779	1578